A140612 -id:A140612 - cp's OEIS frontend

A160053 Duplicate of A140612.

0, 1, 4, 8, 9, 16, 17, 25, 36, 40, 49, 52, 64, 72, 73, 80, 81, 89, 97, 100, 116, 121, 136

Offset: 1

Zak Seidov, May 01 2009

dead

A050795 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in at least one way.

3, 9, 17, 19, 33, 35, 51, 73, 81, 99, 105, 129, 145, 147, 161, 163, 179, 195, 201, 233, 243, 273, 289, 291, 297, 339, 361, 387, 393, 451, 465, 467, 483, 489, 513, 521, 577, 579, 585, 611, 627, 649, 675, 721, 723, 739, 777, 801, 809, 819, 849, 883, 899, 915

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

Analogous solutions exist for the sum of two identical squares z^2-1 = 2.r^2 (e.g. 99^2-1 = 2.70^2). Values of 'z' are the terms in sequence A001541, values of 'r' are the terms in sequence A001542.
Looking at a^2 + b^2 = c^2 - 1 modulo 4, we must have a and b even and c odd. Taking a = 2u, b = 2v and c = 2w - 1 and simplifying, we get u^2 + v^2 = w(w+1). - Franklin T. Adams-Watters, May 19 2008
If n is in this sequence, then so is n^(2^k), for all k >= 0. - Altug Alkan, Apr 13 2016

			E.g. 51^2 - 1 = 10^2 + 50^2 = 22^2 + 46^2 = 34^2 + 38^2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
E.-B. Escott, Query 2521, L'Intermédiaire des Mathématiciens, 10 (1903), 285. [Contains errors]
Index entries for sequences related to sums of squares

Cf. A050796, A050797, A001541, A001542, A001333.

Cf. A140612, A002378.

t={}; Do[i=c=1; While[iJayanta Basu, Jun 01 2013 *)
Select[Range@ 1000, Length[PowersRepresentations[#^2 - 1, 2, 2] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Apr 13 2016 *)

select( {is_A050795(n)=#qfbsolve(Qfb(1,0,1),n^2-1,2)}, [1..999]) \\ M. F. Hasler, Mar 07 2022

from itertools import islice, count
from sympy import factorint
def A050795_gen(startvalue=2): # generator of terms >= startvalue
    for k in count(max(startvalue,2)):
        if all(map(lambda d: d[0] % 4 != 3 or d[1] % 2 == 0, factorint(k**2-1).items())):
            yield k
A050795_list = list(islice(A050795_gen(),20)) # Chai Wah Wu, Mar 07 2022

a(n) = 2*A140612(n) + 1. - Franklin T. Adams-Watters, May 19 2008

{k : A025426(k^2-1)>0}. - R. J. Mathar, Mar 07 2022

A304441 Numbers k such that 8k, 8k+1 and 8k+2 are the sum of two squares; A082982 / 8.

Original entry on oeis.org

0, 1, 2, 9, 10, 18, 29, 36, 45, 65, 72, 73, 100, 101, 136, 137, 144, 153, 164, 200, 208, 218, 225, 234, 245, 281, 288, 289, 298, 324, 325, 353, 416, 424, 441, 450, 514, 522, 541, 578, 640, 648, 666, 676, 738, 757

Offset: 1

M. F. Hasler, May 13 2018

nonn

Numbers n such that n and n+1 are in the sequence: 0, 1, 9, 72, 100, 136, 288, 324, ...: appear to be in A155562, A140612, and A243180, and in A020684 (except for 1), A034024 & A135571 (except for 0, 1).

Cf. A082982, A155562, A243180, A140612, A020684, A034024, A135571.

isA001481(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1
is(n)=isA001481(n) && isA001481(8*n+1) && isA001481(8*n+2) \\ Charles R Greathouse IV, May 17 2018

a(n) = A082982(n) / 8.

A328224 Numbers k such that each of k, k+1, k+2, and k+4 is a sum of two squares.

Original entry on oeis.org

0, 16, 144, 288, 576, 1152, 1600, 2304, 3328, 3600, 4624, 5184, 7056, 8352, 10368, 10656, 10816, 11808, 12112, 12240, 12544, 13120, 13840, 16704, 17424, 19600, 19728, 20736, 20752, 21312, 21904, 22048, 23200, 24480, 24784, 25920, 27792, 28960, 29520, 29824, 30976, 31264, 32400

Offset: 1

Max Alekseyev, Oct 08 2019

nonn

All terms are divisible by 16. - Robert Israel, Oct 10 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001481, A140612, A304441.

Intersection of A082982 and A328223.

[k:k in [0..33000]| forall{k+a: a in [0,1,2,4]|NormEquation(1, k+a) eq true}]; // Marius A. Burtea, Oct 08 2019

ss:=  proc(n) option remember;
  andmap(t -> t[2]::even or t[1] mod 4 <> 3, ifactors(n)[2])
end proc:
select(k -> ss(k) and ss(k+1) and ss(k+2) and ss(k+4), 16*[$0..10^4]); # Robert Israel, Oct 10 2019

ok[n_] := AllTrue[{0,1,2,4}, SquaresR[2, #+n] > 0 &]; Select[ Range[0, 32400], ok] (* Giovanni Resta, Oct 08 2019 *)

A298950 Numbers k such that 5*k - 4 is a square.

Original entry on oeis.org

1, 4, 8, 17, 25, 40, 52, 73, 89, 116, 136, 169, 193, 232, 260, 305, 337, 388, 424, 481, 521, 584, 628, 697, 745, 820, 872, 953, 1009, 1096, 1156, 1249, 1313, 1412, 1480, 1585, 1657, 1768, 1844, 1961, 2041, 2164, 2248, 2377, 2465, 2600, 2692, 2833, 2929, 3076, 3176, 3329, 3433

Offset: 1

Bruno Berselli, Jan 30 2018

a(n) is a member of A140612. Proof: a(n) = n^2 + (n/2-1)^2 for even n, otherwise a(n) = (n-1)^2 + ((n+1)/2)^2; also, a(n) + 1 = (n-1)^2 + (n/2+1)^2 for even n, otherwise a(n) + 1 = n^2 + ((n-3)/2)^2. Therefore, both a(n) and a(n) + 1 belong to A001481.
Primes in sequence are listed in A245042.
Squares in sequence are listed in A081068.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A195162: numbers k such that 5*k + 4 is a square.
Subsequence of A001481, A020668, A036404, A140612.
Cf. A036666, A081068, A106833 (first differences), A245042.

List([1..60], n -> (10*n*(n-1)+(2*n-1)*(-1)^n+9)/8);

[(10*n*(n-1)+(2*n-1)*(-1)^n+9)/8: n in [1..60]];

Table[(10 n (n - 1) + (2 n - 1) (-1)^n + 9)/8, {n, 1, 60}]
LinearRecurrence[{1,2,-2,-1,1},{1,4,8,17,25},60] (* Harvey P. Dale, Sep 16 2022 *)

makelist((10*n*(n-1)+(2*n-1)*(-1)^n+9)/8, n, 1, 60);

Vec((1+x^2)*(1+3*x+x^2)/((1-x)^3*(1+x)^2)+O(x^60))

vector(60, n, nn; (10*n*(n-1)+(2*n-1)*(-1)^n+9)/8)

[(10*n*(n-1)+(2*n-1)*(-1)**n+9)/8 for n in range(1, 60)]

[(10*n*(n-1)+(2*n-1)*(-1)^n+9)/8 for n in (1..60)]

G.f.: x*(1 + x^2)*(1 + 3*x + x^2)/((1 - x)^3*(1 + x)^2).
a(n) = a(1-n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (10*n*(n-1) + (2*n-1)*(-1)^n + 9)/8.
a(n) = A036666(n) + 1.

A328223 Numbers k such that both k and k+4 are sums of two squares.

Original entry on oeis.org

0, 1, 4, 5, 9, 13, 16, 25, 32, 36, 37, 41, 45, 49, 61, 64, 68, 81, 85, 97, 100, 109, 113, 117, 121, 144, 145, 149, 153, 160, 169, 181, 193, 196, 208, 221, 225, 229, 241, 256, 257, 261, 265, 277, 288, 289, 292, 313, 320, 324, 333, 349, 356, 361, 365, 369, 373, 388, 397, 400

Offset: 1

Max Alekseyev, Oct 08 2019

nonn

Subsequence of A001481. Contains A328224 as a subsequence.

Cf. A140612, A082982, A304441.

[k: k in [0..400] | NormEquation(1, k) eq true and NormEquation(1, k+4) eq true]; // Marius A. Burtea, Oct 08 2019

ok[n_] := AllTrue[{0, 4}, SquaresR[2, # + n] > 0 &]; Select[Range[0, 400], ok] (* Giovanni Resta, Oct 08 2019 *)

A333443 Numbers k such that both k and k+1 are sums of two positive squares in 2 or more ways.

Original entry on oeis.org

985, 1585, 1768, 1780, 2249, 2329, 2500, 2929, 3280, 3649, 3977, 4264, 4329, 4705, 4849, 5017, 5044, 5065, 5140, 5161, 5512, 5617, 5625, 6340, 6409, 6697, 7240, 7684, 7785, 7956, 7969, 8020, 8065, 8320, 8584, 8905, 9089, 9265, 9529, 9553, 9593, 9700, 9809

Offset: 1

Mateusz Winiarski, Mar 21 2020

Numbers k such that both k and k+1 belong to A007692.

			985 is a term since 12^2 + 29^2 = 16^2 + 27^2 = 985 and 5^2 + 31^2 = 19^2 + 25^2 = 986.
625 is not a term because 626 cannot be written as the sum of two positive squares in more than one way.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A007692.

Cf. A085323, A140612.

ok[n_] := Length@ IntegerPartitions[n, {2}, Range[Sqrt@ n]^2] >= 2; Select[ Range@ 10000, ok[#] && ok[#+1] &] (* Giovanni Resta, Mar 24 2020 *)

n=100
t=[]
prev=0
A333443=[]
for i in range(1,n+1):
    t.append(i*i)
for j in range(n**2):
    n=0
    for k in t[:j+1]:
        if j-k in t and k<=j-k:
            n=n+1
            if n>1:
                if j-prev==1:
                    A333443.append(j-1)
                prev=j

A374087 a(n) is the number of ways to partition {1,2,...,n} into two sets X and Y such that the sum of the elements of each is a square.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 1, 8, 0, 0, 0, 0, 0, 0, 365, 8, 0, 0, 0, 0, 0, 0, 0, 91514, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 104742767, 0, 0, 0, 6519062, 0, 0, 0, 0, 0, 0, 0, 0, 531168463492, 0, 0, 15329991499, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11530164811834907, 0, 0, 0, 0

Offset: 0

Gonzalo Martínez, Jun 27 2024

nonn

{X, Y} satisfies the property if there exists an integer k such that n*(n+1)/2 - k^2 is a square, where k^2 is the sum of the elements of X and n*(n + 1)/2 - k^2 the sum of the elements of Y.
Note that k can also be zero. If n is A001108(k), i.e., if n*(n + 1)/2 is a perfect square, it suffices to take X = {1, 2, ..., n} and Y = { }.
a(n) > 0 if and only if n is a term of A140612.
Proof: (==>) If n is such that a(n) > 0, then it is possible to partition {1,2,...,n} into two sets, X and Y, whose sums of elements are b^2 and c^2, respectively, for some integers b, c. Then, n*(n + 1)/2 = b^2 + c^2, so, n*(n + 1) = 2*(b^2 + c^2) = (b + c)^2 + (b - c)^2, i.e., n*(n + 1) is a sum of two squares, whence necessarily n and (n + 1) are sums of two squares. Thus, n is in A140612.
(<==) If n is A140612, then n and (n + 1) are sums of two squares, whence it follows that n*(n + 1) is a sum of 2 squares and is also even. Then n*(n + 1)/2 is also a sum of two squares. Then, there exist integers k and m such that n*(n + 1)/2 = k^2 + m^2, so that n*(n + 1)/2 - k^2 = m^2. Therefore, given the set {1,2,...,n}, if we choose X such that the sum of the elements is k^2, it follows that a(n) > 0.

			If n = 4, then the only way is {1}, {2, 3, 4}.
If n = 8, then the only way is { }, {1, 2, 3, 4, 5, 6, 7, 8}.
If n = 9, there are 8 ways, which are shown below:
  {9},  {1, 2, 3, 4, 5, 6, 7, 8}
  {1, 8}, {2, 3, 4, 5, 6, 7, 9}
  {2, 7}, {1, 3, 4, 5, 6, 8, 9}
  {3, 6}, {1, 2, 4, 5, 7, 8, 9}
  {4, 5}, {1, 2, 3, 6, 7, 8, 9}
  {1, 2, 6}, {3, 4, 5, 7, 8, 9}
  {1, 3, 5}, {2, 4, 6, 7, 8, 9}
  {2, 3, 4}, {1, 5, 6, 7, 8, 9}
In each of the 8 cases, the sum of the elements of the subsets are 9 and 36, respectively.
If n = 25, there are 91514 ways. Some examples with sums different from each other:
  {1}, {2, 3, ..., 25}, where the sums are 1^2 and 18^2, respectively.
  {1, 2, 3, 4, 5, 6, 7, 8}, {9, 10, 11, ..., 25}, where the sums are 6^2 and 17^2.
  X = {6, 22, 23, 24, 25}, Y = {1, 2, ..., 25} - X, whose sums are 10^2 and 15^2.

Cf. A000217, A000290, A002849, A140612, A001108.

a(36)-a(68) from Alois P. Heinz, Jun 29 2024

cp's OEIS Frontend

A160053 Duplicate of A140612.

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A050795 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in at least one way.

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A304441 Numbers k such that 8k, 8k+1 and 8k+2 are the sum of two squares; A082982 / 8.

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A328224 Numbers k such that each of k, k+1, k+2, and k+4 is a sum of two squares.

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Magma

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A298950 Numbers k such that 5*k - 4 is a square.

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Maxima

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A328223 Numbers k such that both k and k+4 are sums of two squares.

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Magma

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A333443 Numbers k such that both k and k+1 are sums of two positive squares in 2 or more ways.

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A374087 a(n) is the number of ways to partition {1,2,...,n} into two sets X and Y such that the sum of the elements of each is a square.

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